Derivative Calculator
Differentiate a function and see which rule does the work at every term — power, product, quotient and chain — plus the value, tangent line and second derivative.
Derivative Calculator: with the default inputs, derivative f′(x) is 3 · x^2 − 10 · x + 2.
Use ^ for powers. sin, cos, tan, exp, log (natural), ln, sqrt and the rest are available.
The slope of the curve at this value of x. Solve for it to find where the slope hits a target.
- f′ at the point
- -6The slope of the curve there.
- f at the point
- -16
- Tangent line
- y = −6x − 4
- Second derivative f″(x)
- 6 · x − 10
- f″ at the point
- 2Positive means the curve is bending upward there.
- Rules used
- Power rule; Constant multiple rule; Constant rule
Assumptions
- Differentiation is symbolic and exact; only the evaluation at a point is floating point.
- One variable at a time — no implicit differentiation or partial derivatives.
- log is the natural logarithm (ln is accepted as a synonym); log10 and log2 are available separately.
- The term-by-term assembly is checked against mathjs's own whole-expression derivative at several points before it is displayed; if they ever disagree, mathjs's version is shown instead.
- Very large expressions are rejected rather than differentiated slowly, and an unusually bulky derivative is shown unsimplified with a note.
| Term | Rule that applies | Its derivative |
|---|---|---|
| x^3 | Power rule — d/d· uⁿ = n · uⁿ⁻¹ · u′ | 3 · x^2 |
| −5 x^2 | Constant multiple rule — the constant comes along unchanged | −10 · x |
| 2 x | Constant multiple rule — the constant comes along unchanged | 2 |
| −8 | Constant rule — the derivative of a constant is 0 | 0 |
The sum rule lets each term be differentiated on its own; adding the right-hand column back together gives f′.
How this is worked out
The formula
Constant: d/dx c = 0 Power: d/dx xⁿ = n·xⁿ⁻¹ Constant multiple: d/dx c·f = c·f′ Sum: d/dx (f + g) = f′ + g′ Product: d/dx (f·g) = f′g + fg′ Quotient: d/dx (f ÷ g) = (f′g − fg′) ÷ g² Chain: d/dx f(g(x)) = f′(g(x))·g′(x) Tangent line at a: y = f(a) + f′(a)(x − a)
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Function f(x)
- Use ^ for powers. sin, cos, tan, exp, log (natural), ln, sqrt and the rest are available.up to 120 characters · defaults to "x^3 - 5x^2 + 2x - 8"
- Evaluate the derivative at
- The slope of the curve at this value of x. Solve for it to find where the slope hits a target.defaults to 2
- Differentiate with respect to(under More options)
- Free text.up to 1 characters · defaults to "x"
What you get back
- Derivative f′(x)main answer
- f′ at the point
- The slope of the curve there.
- f at the point
- Tangent line
- Second derivative f″(x)
- f″ at the point
- Positive means the curve is bending upward there.
- Rules used
What this assumes
- Differentiation is symbolic and exact; only the evaluation at a point is floating point.
- One variable at a time — no implicit differentiation or partial derivatives.
- log is the natural logarithm (ln is accepted as a synonym); log10 and log2 are available separately.
- The term-by-term assembly is checked against mathjs's own whole-expression derivative at several points before it is displayed; if they ever disagree, mathjs's version is shown instead.
- Very large expressions are rejected rather than differentiated slowly, and an unusually bulky derivative is shown unsimplified with a note.
About this calculator
The derivative of a function is its slope — how fast the output changes when the input moves. This differentiates the function you type, names the rule that does the work at each term, evaluates the slope at a point, and gives you the tangent line there.
How to use it
Type the function with ^ for powers: x^3 - 5x^2 + 2x - 8, x^2*sin(x), (x^2+1)/(x-3), sqrt(x^2+1), exp(3x). Implicit multiplication works, so 5x^2 and 5*x^2 are the same thing. log is the natural logarithm and ln is accepted as a synonym; log10 and log2 are there too. Set the point to the x-value you care about, or use Solve for to find the x where the slope hits a target — solving for a slope of 0 finds the maxima, minima and saddle points.
Reading the term-by-term table
Every derivative in a first calculus course is the same handful of rules applied in the right order, and the table under the results shows which one each term needs:
- Power rule — xⁿ becomes n·xⁿ⁻¹. The workhorse for polynomials.
- Constant multiple — a number in front just comes along: (5x²)′ = 5·(x²)′ = 10x.
- Product rule — (uv)′ = u′v + uv′. Needed when both factors contain x; if one is a plain number it is the constant multiple rule instead, which is where most marks are lost.
- Quotient rule — (u/v)′ = (u′v − uv′)/v². Dividing by a constant does not need it.
- Chain rule — the derivative of the outside times the derivative of the inside. sin(x²) becomes cos(x²)·2x, not cos(2x).
What the other numbers tell you
f′ at the point is the slope of the curve there: positive means rising, negative means falling, zero means a horizontal tangent — a maximum, a minimum, or a point of inflection.
The tangent line is the straight line that touches the curve at that point, and it is the best straight-line approximation to the function nearby. It is also what the chart draws through the curve, so you can see the slope rather than just read it.
The second derivative is the derivative of the derivative: it measures curvature. Positive means the curve bends upward (concave up), negative means it bends downward. Combined with f′ = 0 that is the second-derivative test — f′(a) = 0 with f″(a) > 0 is a local minimum, f″(a) < 0 a local maximum.
Limits worth knowing
This does symbolic differentiation, not numerical approximation, so the answer is exact. It handles one variable at a time, and it does not do implicit differentiation, partial derivatives or integrals — mathematically, integration has no equivalent of the chain rule to run backwards, which is exactly why a derivative calculator can always succeed and an antiderivative calculator cannot.
Frequently asked questions
▸What is the derivative of x³ − 5x² + 2x − 8?
3x² − 10x + 2. Each term follows the power rule: x³ → 3x², −5x² → −10x, 2x → 2, and the constant −8 differentiates to 0.
▸When do I need the product rule instead of the constant multiple rule?
Only when both factors contain the variable. (5x²)′ is 5 × (x²)′ = 10x — no product rule. (x²·sin x)′ does need it: 2x·sin x + x²·cos x.
▸How does the chain rule work?
Differentiate the outside function, leave the inside alone, then multiply by the derivative of the inside. sin(x²) becomes cos(x²) × 2x. The table under the results says 'with the chain rule' whenever it is applied.
▸What does the second derivative tell me?
Curvature. Positive means the curve bends upward, negative downward. Where f′ = 0, a positive f″ marks a local minimum and a negative f″ a local maximum.
▸Can it find integrals too?
Not symbolically — and nothing that claims to should be trusted without checking. The integral calculator on this site does definite integrals numerically with Simpson's rule and says so plainly.
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Related calculators
The questions people ask next to a derivative.
Definite integrals by composite Simpson's rule, with an error estimate, a convergence table and the area drawn. Numerical — it does not produce an antiderivative.
Solve any linear or quadratic equation in one variable with the rearrangement shown line by line — discriminant, exact roots, vertex, factored form and a check.
Slope, y-intercept and the equation of the line through two points, plus distance, midpoint, angle and the perpendicular slope — with rise-over-run shown.
Solve ax² + bx + c = 0 with the quadratic formula — real or complex roots, discriminant, vertex, axis of symmetry, factored form and a graph of the parabola.
Type a whole expression and get an exact answer: brackets, powers, roots, logs, trigonometry in degrees or radians, and the order of operations applied properly.
Raise any base to any power — negative, fractional or huge — with the expanded multiplication, scientific notation and the rule that applies shown.